The term adiabatic lapse rate used by meteorologists is the
decrease in temperature with height that results from the adiabatic
expansion of an air mass as it is pushed up a mountain by the wind.
Similarly, the wind coming down the mountain slope warms up. This
adiabatic expansion is represented by P₀, V₀,
T₀ P, V, T
where ΔS = 0 and P₀, V₀, T₀
represents the sea level conditions. The calculation of the entropy
change can be carried out in two steps: the first at constant
pressure P₀, V₀, T₀
P_0, V*, T, the second at constant temperature
P₀, V*, T₀ P, V,
T.

a. Calculate ΔS1 and ΔS2, the change in entropy for the two
steps above.

b. Since ΔS = ΔS1 + ΔS2, what is the expression for ΔT/Δh. To
derive this use the expression for the change in pressure with
height in the atmosphere from Problem Set 2 (P_i =
P_i,0 exp(-M_igh / RT).

c. If the temperature at the foot of a 14,000-foot mountain is
25 °C, what temperature would you expect at the summit from this
adiabatic lapse rate? For this calculation the molar mass of air
can be taken to be M = 29 g mol^-1 and its heat capacity
can be taken as 29.1 J K^-1mol^-1.